91Ó°ÊÓ

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int \frac{3 x}{x+2} d x, \text { with } u=x+2 $$

Use the trapezoidal rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. $$ \int_{0}^{2} \sqrt{x} d x, n=4 $$

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int x \sec ^{2} x d x $$

Use the Table of Integrals to compute each integral after manipulating the integrand in a suitable way. $$ \int \frac{1}{\sqrt{16-9 x^{2}}} d x $$

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{2}+1}{3 x+1} $$

Use the trapezoidal rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. $$ \int_{1}^{2} \frac{1}{x} d x, n=5 $$

Compute the Taylor polynomial of degree \(n\) about \(x=0\) for each function and compare the value of the function at the indicated point with the value of the corresponding Taylor polynomial. $$ f(x)=\ln (1+x), n=3, x=0.1 $$

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int \frac{x+1}{5-x} d x, \text { with } u=5-x $$

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int x \csc ^{2} x d x $$

All the integrals in problem are improper and converge. Explain in each case why the integral is improper, and evaluate each integral. $$ \int_{0}^{2} \frac{d x}{(x-1)^{2 / 5}} $$